from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8712, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([33,0,55,42]))
pari: [g,chi] = znchar(Mod(23,8712))
Basic properties
| Modulus: | \(8712\) | |
| Conductor: | \(4356\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(66\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{4356}(23,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8712.dh
\(\chi_{8712}(23,\cdot)\) \(\chi_{8712}(551,\cdot)\) \(\chi_{8712}(815,\cdot)\) \(\chi_{8712}(1343,\cdot)\) \(\chi_{8712}(1607,\cdot)\) \(\chi_{8712}(2135,\cdot)\) \(\chi_{8712}(2399,\cdot)\) \(\chi_{8712}(2927,\cdot)\) \(\chi_{8712}(3191,\cdot)\) \(\chi_{8712}(3719,\cdot)\) \(\chi_{8712}(3983,\cdot)\) \(\chi_{8712}(4511,\cdot)\) \(\chi_{8712}(4775,\cdot)\) \(\chi_{8712}(5303,\cdot)\) \(\chi_{8712}(6095,\cdot)\) \(\chi_{8712}(6359,\cdot)\) \(\chi_{8712}(6887,\cdot)\) \(\chi_{8712}(7151,\cdot)\) \(\chi_{8712}(7679,\cdot)\) \(\chi_{8712}(7943,\cdot)\)
sage: chi.galois_orbit()
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Related number fields
| Field of values: | \(\Q(\zeta_{33})\) |
| Fixed field: | Number field defined by a degree 66 polynomial |
Values on generators
\((6535,4357,1937,5689)\) → \((-1,1,e\left(\frac{5}{6}\right),e\left(\frac{7}{11}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 8712 }(23, a) \) | \(1\) | \(1\) | \(e\left(\frac{17}{66}\right)\) | \(e\left(\frac{19}{66}\right)\) | \(e\left(\frac{31}{33}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{7}{33}\right)\) | \(e\left(\frac{17}{33}\right)\) | \(e\left(\frac{43}{66}\right)\) | \(e\left(\frac{59}{66}\right)\) | \(e\left(\frac{6}{11}\right)\) |
sage: chi.jacobi_sum(n)